Normalizer of maximal abelian subgroups of \(\mathrm{GL}(n)\) and general hypergeometric functions. (English) Zbl 0848.33009

Kimura, Hironobu; Koitabashi, Toshiyuki

Kumamoto J. Math. 9, 13-43 (1996).

The generalized confluent hypergeometric systems on the Grassmannian have been formulated in the earlier papers of I. M. Gel’fand, V. S. Retakh, and V. V. Serganova [Generalized Airy functions, Schubert cells, and Jordan groups, Sov. Math., Dokl. 37, No. 1, 8–12 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 1, 17–21 (1988; Zbl 0699.33012)] and H. Kimura, Y. Haraoka, and K. Takano [Generalized confluent hypergeometric functions, Proc. Jap. Acad., Ser. A 68, 290–295 (1992; Zbl 0773.33004)]. The paper clarifies the symmetry of the confluent hypergeometric systems in terms of the action of the normalizer of maximal abelian subgroups of \(\mathrm{GL}(n)\). The symmetry particularly reveals several transformation formulas for the classical confluent hypergeometric functions.

Reviewer: T.Sasaki (Kobe)

MSC:

33C80	Connections of hypergeometric functions with groups and algebras, and related topics
33C70	Other hypergeometric functions and integrals in several variables
33C65	Appell, Horn and Lauricella functions

Keywords:

confluent hypergeomemetric function; symmetry of confluent hypergeometric systems